Let's say I have an online photo gallery. Well the usual way to do this is store a URL to the images in a table. This, however, seems like a poor separation, as the website is a front-end and the table is a back-end, and you have a server referring to something in a client.

Instead, why not store the images as separate files on the DB server itself, and use a view/query to read the image from disk and return it to the client as a binary blob. This way, any front-end has access to the images, not just that specific web server.

What are the pitfalls of this approach?

That is exactly what SQL Server Filestream does:
FILESTREAM Overview

The problem with a roll-your-own approach is that you won't get native support for transactions, integrity, backup and recovery. You still have to manage the file system data yourself. Native support like Filestream means that those things are taken care of for you even for the data hosted on the filesystem.

I'm running PostgreSQL.